D ifferential E quations & A pplications Volume 8, Number 4 (2016), 459–470 doi:10.7153/dea-08-26 COUPLED SYSTEMS OF FRACTIONAL ∇ –DIFFERENCE BOUNDARY VALUE PROBLEMS YOUSEF GHOLAMI AND KAZEM GHANBARI (Communicated by Chris Goodrich) Abstract. In this paper, we study the existence of solutions for a coupled system of two-point fractional ∇ -difference boundary value problems of the form  ∇ α a + u(t ) ∇ β a + v(t )  +  f (t , v(t ) g(t , u(t ))  = 0,  u(a + 1) u(b + 1)  =  0 0  =  v(a + 1) v(b + 1)  , where 1 < α , β  2, t ∈ [a + 2, b + 1] N = {a + 2, a + 3, ..., b, b + 1}, a, b ∈ Z such that a  0, b  3 and the functions f , g : [a + 2, b + 1] N × R → R are continuous. Our analysis relies on the Green functions and the nonlinear alternative of Leray-Schauder and Krasnosel´ skii-Zabreiko fixed point theorems. At the end we give some numerical examples to illustrate the main results. 1. Introduction The theory of fractional calculus basically acts on the differential operators as D α t ≡ d α /dt α with arbitrary order α ∈ R , that generalize the integer order integration and differentiation. In recent decades, it has been illustrated that many systems ap- peared in science and engineering can be simulated by fractional derivatives rather than integer ones [11, 12]. This is why we are interested to the study of the various fractional based approaches related to the both theoretical and computational sciences interacted with mathematics. Despite the boom of developments in fractional differential equations, the ap- proach of the fractional difference equations have been included to the collection of some elementary analysis of fractional discrete boundary value problems in the early last decade. In this way one can suggest the pioneering works of F. Atici and P. Eloe [2, 3, 4, 5], C. Goodrich [7], C. Goodrich, A. C. Peterson [8], Y. Gholami and K. Ghan- bari [9], R. A. C. Ferreira [6]. F. Atici and P. Eloe in [3], studied the two-point fractional Δ -difference boundary value problem  -Δ ν y(t )= f (t + ν - 1, y(t + ν - 1)), t = 1, 2, ..., b + 1, y(ν - 2)= 0, y(ν + b + 1)= 0, (1.1) Mathematics subject classification (2010): 34A08, 39A12, 34A12, 34B15, 39A10. Keywords and phrases: discrete fractional calculus, boundary value problems, fixed point theorem, existence of solutions. c  , Zagreb Paper DEA-08-26 459